3 large skinless, boneless chicken breasts (about two pounds)
3 tablespoons extra-virgin olive oil, or more if needed
12 large shiitake mushrooms, stems trimmed
All-purpose flour, for dredging
½ teaspoon salt, or more to taste
3 tablespoons unsalted butter, softened
1 cup tomato sauce
1 cup dry white wine
6 fresh sage leaves
¾ cup freshly grated Grana Padano
1 cup chicken stock or vegetable stock, if needed
Preheat the oven to 425 degrees. Cut the chicken breasts in half on a bias. Trim off any bits of fat, skin, or tendon. Flatten each breast half with a mallet to an even thickness of about ¾ inch.
Heat three tablespoons of the olive oil in a large skillet over medium heat. Toss the mushroom caps in flour to coat well, pat off any excess, and lay them in the pan. Let the pieces caramelize slowly, for about four minutes. Turn when the edges are nicely browned on the underside; fry for about two minutes on the second side. Remove them to a paper-towel-lined plate, and season with salt.
When the mushrooms are done, add one tablespoon of the butter, and reduce the heat to medium low. Salt the chicken pieces, and dredge in flour, tapping off the excess. When the butter is just beginning to sizzle, lay the chicken cutlets in the pan.
Cook them gently until very lightly colored, with no browning, about two minutes, then flip them and cook the other side, another two minutes.
Maintain the gentle cooking while you assemble the gratinate. Spread a heaping tablespoon of tomato sauce on top of each cutlet. Add a third of the Grana Padano. Arrange the mushroom caps on top of the sauced chicken, using two mushroom caps per piece of chicken. Top with another third of the Grana Padano.
Raise the heat a bit, and begin to develop the sauce: Drop the remaining butter, in small pieces, in between the layered cutlets. After a few seconds, pour in ⅔ cup of the wine around the cutlets, and let it heat briefly, 10 to 20 seconds. Spoon the rest of the cup of tomato sauce into the pan (not on the chicken). Bring it to a simmer, and then drop the sage into the sauce, distributing it all around the pan. Give the pan a gentle shake or two, to mix and emulsify the sauce ingredients. Add the remaining wine (or stock, or water, if necessary, so the sauce comes almost to the top of the chicken), plus the last of the cheese, bring it to a simmer, and place the chicken in the oven.
Bake, shaking the pan once or twice, until the chicken is cooked through and the top is crusty and golden, about 15 to 20 minutes, adding a little more stock or water if the sauce gets too thick. Remove sage. To serve, place on a plate and spoon sauce around it.